To solve the problem of finding the maximum extension in the spring connecting two blocks of different masses, we can use the principles of conservation of momentum and energy. Let's break down the steps involved in this process.
Understanding the System
We have two blocks: one with a mass of 3 kg moving at a velocity of 1 m/s, and the other with a mass of 6 kg moving at a velocity of 2 m/s. They are moving in opposite directions, which means we need to consider their velocities as vectors. The spring connecting them has a spring constant (k) of 200 N/m and is initially unstretched.
Step 1: Calculate Initial Momentum
The total momentum of the system before the spring starts to stretch can be calculated using the formula:
- Momentum = mass × velocity
For the 3 kg block moving to the right (we'll consider this as positive):
- Momentum of 3 kg block = 3 kg × 1 m/s = 3 kg·m/s
For the 6 kg block moving to the left (considered negative):
- Momentum of 6 kg block = 6 kg × (-2 m/s) = -12 kg·m/s
Now, the total initial momentum (P_initial) is:
- P_initial = 3 kg·m/s - 12 kg·m/s = -9 kg·m/s
Step 2: Conservation of Momentum
When the spring is stretched, the blocks will momentarily come to a stop before moving back towards each other. At this point, the total momentum of the system must still equal the initial momentum:
- P_final = 0 (since they momentarily stop)
This means that the momentum lost by the faster block (6 kg) will be equal to the momentum gained by the slower block (3 kg) when the spring is maximally extended.
Step 3: Calculate Maximum Extension Using Energy Conservation
As the blocks move apart, kinetic energy is converted into potential energy stored in the spring. The initial kinetic energy (KE_initial) of the system can be calculated as follows:
- KE_initial = (1/2) * m1 * v1² + (1/2) * m2 * v2²
Substituting the values:
- KE_initial = (1/2) * 3 kg * (1 m/s)² + (1/2) * 6 kg * (2 m/s)²
- KE_initial = (1/2) * 3 * 1 + (1/2) * 6 * 4
- KE_initial = 1.5 + 12 = 13.5 J
Step 4: Potential Energy in the Spring
At maximum extension (x), all the kinetic energy will be converted into potential energy (PE) stored in the spring:
- PE_spring = (1/2) * k * x²
Setting the initial kinetic energy equal to the potential energy at maximum extension:
- 13.5 J = (1/2) * 200 N/m * x²
Now, solving for x:
- 13.5 = 100 * x²
- x² = 0.135
- x = √0.135 ≈ 0.367 m
Final Result
The maximum extension in the spring when the two blocks are moving apart is approximately 0.367 meters or 36.7 centimeters. This demonstrates how energy conservation principles can be applied to analyze the behavior of connected systems in motion.
